1. A new definition for the dynamics
G. Bertrand
Laboratoire A2SI, ESIEE
Institut Gaspard Monge – UMR UMLV/ESIEE/CNRS
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2. A discrete approach Let G = (V,E) be an (undirected) graph.
We denote by Func (V) the family composed of all maps from V to Z.
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3. Pass value
Let F be in Func (V). If п is a path, we set F(п) = Max{F(x); x  п}.
Let x, y in V. We set F(x,y) = Min {F(п); п  п(x,y)}, F(x,y) is the pass value between x and y.
Let X and Y be two subsets of V. We set F(X,Y) = Min{F(x,y); x  X and y  Y}.
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4. Pass value 40 40 40 40 40 40 40 40 40 40 40 40 40 40 1 1 2 3 10 5 25 5 4 4 4 40 40 1 2 8 6 5 5 20 3 2 3 40 40 3 3 2 3 10 6 6 6 22 2 3 40 40 6 6 40 6 11 11 11 25 4 4 4 40 40 40 35 10 30 15 15 15 35 31 36 10 40 40 10 8 5 10 32 33 34 10 10 15 38 40 40 8 5 1 1 15 40 10 6 3 15 20 40 40 10 8 5 10 15 35 15 6 6 15 35 40 40 40 40 40 40 40 40 40 40 40 40 40 40
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5. Pass value 40 40 40 40 40 40 40 40 40 40 40 40 40 40 1 1 2 3 10 5 25 5 4 4 4 40 40 1 2 8 6 5 5 20 3 2 3 40 40 3 3 2 3 10 6 6 6 22 2 3 40 40 6 6 40 6 11 11 11 25 4 4 4 40 40 40 35 10 30 15 15 15 35 31 36 10 40 40 10 8 5 10 32 33 34 10 10 15 38 40 40 8 5 1 1 15 40 10 6 3 15 20 40 40 10 8 5 10 15 35 15 6 6 15 35 40 40 40 40 40 40 40 40 40 40 40 40 40 40
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6. Pass value 40 40 40 40 40 40 40 40 40 40 40 40 40 40 1 1 2 3 10 5 25 5 4 4 4 40 40 1 2 8 6 5 5 20 3 2 3 40 40 3 3 2 3 10 6 6 6 22 2 3 40 40 6 6 40 6 11 11 11 25 4 4 4 40 40 40 35 10 30 15 15 15 35 31 36 10 40 40 10 8 5 10 32 33 34 10 10 15 38 40 40 8 5 1 1 15 40 10 6 3 15 20 40 40 10 8 5 10 15 35 15 6 6 15 35 40 40 40 40 40 40 40 40 40 40 40 40 40 40
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7. Pass value 40 40 40 40 40 40 40 40 40 40 40 40 40 40 1 1 2 3 10 5 25 5 4 4 4 40 40 1 2 8 6 5 5 20 3 2 3 40 40 3 3 2 3 10 6 6 6 22 2 3 40 40 6 6 40 6 11 11 11 25 4 4 4 40 40 40 35 10 30 15 15 15 35 31 36 10 40 40 10 8 5 10 32 33 34 10 10 15 38 40 40 8 5 1 1 15 40 10 6 3 15 20 40 40 10 8 5 10 15 35 15 6 6 15 35 40 40 40 40 40 40 40 40 40 40 40 40 40 40
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F(X,Y) = 31

8. Dynamics (M. Grimaud,1992)
Let X be a minimum for F Let G(X) be the number such that: i) if X = Xmin, then G(X) = infinity; ii) otherwise, G(X) = Min {F(X,Y); for all minima Y such that F(Y) < F(X)}.
The dynamics of a minimum X is the number Dyn(X) = G(X) – F(X)
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9. Dynamics
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10. Dynamics ∞
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11. Dynamics ∞
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12. Dynamics ∞
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13. Dynamics ∞
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14. Dynamics ∞
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15. Dynamics ∞
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16. k-Separation
Let F be in Func (V) and let x and y be in V We say that x and y are separated (for F) if F(x,y) > Max{F(x),F(y)}. We say that x and y are k-separated (for F) if x and y are separated and F(x,y) = k.
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17. x and y are not separated
k-separation
x and y are not separated 40 40 40 40 40 40 40 40 40 40 40 40 40 40 1 1 2 3 10 5 25 5 4 4 4 40 40 1 2 8 6 5 5 20 3 2 3 40 x y 40 3 3 2 3 10 6 6 6 22 2 3 40 40 6 6 40 6 11 11 11 25 4 4 4 40 40 40 35 10 30 15 15 15 35 31 36 10 40 40 10 8 5 10 32 33 34 10 10 15 38 40 40 8 5 1 1 15 40 10 6 3 15 20 40 40 10 8 5 10 15 35 15 6 6 15 35 40 40 40 40 40 40 40 40 40 40 40 40 40 40
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18. k-separation y x x and y are 20-separated ISMM 2005 40 40 40 40 40 401 1 2 3 10 5 25 5 4 4 4 40 x y 40 1 2 8 6 5 5 20 3 2 3 40 40 3 3 2 3 10 6 6 6 22 2 3 40 40 6 6 40 6 11 11 11 25 4 4 4 40 40 40 35 10 30 15 15 15 35 31 36 10 40 40 10 8 5 10 32 33 34 10 10 15 38 40 40 8 5 1 1 15 40 10 6 3 15 20 40 40 10 8 5 10 15 35 15 6 6 15 35 40 40 40 40 40 40 40 40 40 40 40 40 40 40
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19. Separation
Let F and G be in Func (V) such that G  F. We say that G is a separation of F if, for all x,y in V, if x and y are k-separated for F, then x and y are k-separated for G.
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20. Separation F
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21. Separation F
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22. Separation F K
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23. Separation F
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24. Separation F K
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25. Separation F K
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G is a separation of F G

26. Dynamics and separation
Let G ≤ F (G being a minima extension of F)
If G is a separation of F, then the dynamics of a minimum of G is the same than the dynamics of the corresponding minimum of F
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27. Dynamics and separation
Let G ≤ F (G being a minima extension of F).
If G is a separation of F, then the dynamics of a minimum of G is the same than the dynamics of the corresponding minimum of F.
The converse is not true
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28. Dynamics: counter-exampleF ∞
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29. Dynamics: counter-exampleG ∞
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30. Ordered minima
Let F be in F (V). A minima ordering (for F) is a strict total order relation < on the minima of F.
Let X be a minimum for F. The pass value of X for (F,<) is the number F(X,<) such that: i) if X = Xmin, then F(X,<) = infinity; ii) otherwise, F(X,<) = Min {F(X,Y); for all minima Y such that Y < X}.
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31. Ordered minima 40 40 40 40 40 40 40 40 40 40 40 40 40 40 1 1 2 3 10 5 25 5 4 4 4 40 40 1 2 8 6 5 5 20 3 2 3 40 40 3 3 2 3 10 6 6 6 22 2 3 40 40 6 6 40 6 11 11 11 25 4 4 4 40 40 40 35 10 30 15 15 15 35 31 36 10 40 40 10 8 5 10 32 33 34 10 10 15 38 40 40 8 5 1 1 15 40 10 6 3 15 20 40 40 10 8 5 10 15 35 15 6 6 15 35 40 40 40 40 40 40 40 40 40 40 40 40 40 40
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32. Ordered minima F(.,<)=8 5 3 2 F(.,<)=20 F(.,<)=30 1 440 40 40 40 40 40 40 40 40 40 40 40 40 5 40 3 1 1 2 3 10 5 25 5 4 4 4 40 2 40 1 2 8 6 5 5 20 3 2 3 40
F(.,<)=20 40 3 3 2 3 10 6 6 6 22 2 3
F(.,<)=30 40 40 6 6 40 6 11 11 11 25 4 4 4 40 40 40 35 10 30 15 15 15 35 31 36 10 40 40 10 8 5 10 32 33 34 10 10 15 38 40 1 4 40 8 5 1 1 15 40 10 6 3 15 20 40 40 10 8 5 10 15 35 15 6 6 15 35 40
F(.,<)=infty
F(.,<)=31 40 40 40 40 40 40 40 40 40 40 40 40 40
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33. Ordered dynamics
The notion of ordered pass values leads to a new definition of the dynamics of a minimum: Dyn(X; F, <) = F(X, <) – F(X)
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34. Ordered minima Dyn(.,<)=8-5 5 3 2 Dyn(.,<)=20-0 Dyn(.,<)=30-240 40 40 40 40 40 40 40 40 40 40 40 40 5 40 3 1 1 2 3 10 5 25 5 4 4 4 40 2 40 1 2 8 6 5 5 20 3 2 3 40 40 3 3 2 3 10 6 6 6 22 2 3 40
Dyn(.,<)=20-0 40 6 6 40 6 11 11 11 25 4 4 4 40
Dyn(.,<)=30-2 40 40 35 10 30 15 15 15 35 31 36 10 40 40 10 8 5 10 32 33 34 10 10 15 38 40 1 4 40 8 5 1 1 15 40 10 6 3 15 20 40 40 10 8 5 10 15 35 15 6 6 15 35 40
Dyn(.,<)=31-3 40 40 40 40 40 40 40 40 40 40 40 40 40
Dyn(.,<)=infty
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35. Theorem (ordered dynamics and separation)
Let G ≤ F (G being a minima extension of F).
Let < be a minima ordering for F The map G is a separation of F if and only if, for each minimum X for F, we have Dyn(X; F, <) = Dyn(X; G, <) .
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36. Dynamics: counter-example∞
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37. Ordered minima F
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38. Ordered minima F 2 3
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39. Ordered minima ∞ F 2 3
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40. Ordered minima ∞ F 2 3
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41. Ordered minima ∞ G 2 3
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42. Ordered minima F 1 2
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43. Ordered minima ∞ F 1 2
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44. Ordered minima ∞ F 1 2
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45. Ordered minima ∞ G 1 2
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46. Remark
If all the minima of a function F are distinct and if the ordering of the minima of F is made according to the altitudes of the minima of F, then the ordered dynamics of a minimum is equal to the unordered dynamics of this minimum.
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47. A tree associated to F and <40 40 40 40 40 40 40 40 40 40 40 40 40 5 40 3 1 1 2 3 10 5 25 5 4 4 4 40 2 40 1 2 8 6 5 5 20 3 2 3 40
F(.,<)=20 40 3 3 2 3 10 6 6 6 22 2 3
F(.,<)=30 40 40 6 6 40 6 11 11 11 25 4 4 4 40 40 40 35 10 30 15 15 15 35 31 36 10 40 40 10 8 5 10 32 33 34 10 10 15 38 40 1 4 40 8 5 1 1 15 40 10 6 3 15 20 40 40 10 8 5 10 15 35 15 6 6 15 35 40
F(.,<)=0
F(.,<)=31 40 40 40 40 40 40 40 40 40 40 40 40 40
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48. Theorem (minimum spanning tree)
Let F be in F (V) and let < be a minima ordering for F. Let T be a tree associated to F and <. Let G’ be the complete graph the vertices of which are the minima of F, an edge being labeled by the corresponding pass value. The tree T is a minimum spanning tree of G’.
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49. Conclusion
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Dyn > 22

50. Conclusion
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=> Ordering the minima with arbitary criteria

51. Conclusion Preservation of the dynamics
Equivalence between :
Preservation of the dynamics
Preservation of the contrast (separation)
Preservation of an optimal spanning tree
Preservation of the crests (topological watersheds)
Preservation of the components of the cross-sections (extension)
Preservation of the component tree
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52. Topological watershed
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53. Conclusion Preservation of the dynamics
Equivalence between :
Preservation of the dynamics
Preservation of the contrast (separation)
Preservation of an optimal spanning tree
Preservation of the crests (topological watersheds)
Preservation of the components of the cross-sections (extension)
Preservation of the component tree
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54. Components of the cross-sections
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55. Components of the cross-sections
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56. Components of the cross-sections
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57. Conclusion Preservation of the dynamics
Equivalence between :
Preservation of the dynamics
Preservation of the contrast (separation)
Preservation of an optimal spanning tree
Preservation of the crests (topological watersheds)
Preservation of the components of the cross-sections (extension)
Preservation of the component tree
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58. Components of the cross-sections
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59. Components of the cross-sections
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60. Components of the cross-sections
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61. Conclusion Preservation of the dynamics
Equivalence between :
Preservation of the dynamics
Preservation of the contrast (separation)
Preservation of an optimal spanning tree
Preservation of the crests (topological watersheds)
Preservation of the components of the cross-sections (extension)
Preservation of the component tree
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62. Thank you for your attention
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63. Theorem (reconstruction from ordered pass values)
Let F be in F (V) and let < be a minima ordering for F. Let T be a tree associated to F(X,<). The pass values between all minima of F may be reconstructed from T.
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64. Separation (sets)
Let X be a subset of E and let x, y be in X. We say that x and y are separated for X if there is no path from x to y in X.
Let X, Y be subsets of E such that Y is a subset of X. We say that Y is a separation of X if any x and y in X which are separated for X, are separated for Y.
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65. A subset X
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66. A separation
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67. Separation (maps)
We denote by Func (V) the family composed of all maps from V to Z.
Let F  Func (V), we set Fk = {x  V; F(x)  k}, Fk is the cross-section of F at level k
Let F and G be both in Func(V) and such that G ≤ F. We say that G is a separation of F if, for any k, G[k] is a separation of F[k].
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68. Strong separation F
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69. Discrete sets and destructible points
Let G = (V,E) be a (undirected) graph and let X be a subset of V.
We say that a point x  X is destructible for X if x is adjacent to exactly one connected component of X.
M. Couprie and G. Bertrand (1997)
Watersheds
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70. Theorem (restriction to minima)
Let F and G be in F (V) such that G  F. The map G is a separation of F if and only if, for all distinct minima X,Y for F, F(X,Y) = G(X,Y).
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71. Theorem (strong separation)
Let F and G be in F (V) such that G  F. The map G is a strong separation of F if and only if G is a W-thinning of F.
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72. Theorem (confluence)
Let G be a W-thinning of F. If H is a W-thinning of F such that H >= G, then G is a W-thinning of H
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73. Strong separation F
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G is a strong separation of F G

74. Strong separation F
destructible points may be lowered with an arbitrary order
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75. Theorem (restriction to minima)
Let F and G be in F (V) such that G  F. The map G is a separation of F if and only if, for all distinct minima X,Y for F, F(X,Y) = G(X,Y).
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76. Theorem (restriction to minima)
Let F and G be in F (V) such that G  F. The map G is a separation of F if and only if, for all distinct minima X,Y for F, F(X,Y) = G(X,Y).
Is it possible to reduce the amount of information necessary to ‘‘encode’’ the  topology of a W-thinning?
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77. Ordered minima
Let F be in F (V). A minima ordering (for F) is a strict total order relation < on the minima of F.
Let X be a minimum for F. The pass value of X for (F,<) is the number F(X,<) such that: i) if X = Xmin, then F(X,<) = infinity; ii) otherwise, F(X,<) = Min {F(X,Y); for all minima Y such that Y < X}.
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78. Theorem (ordered minima)
Let F and G be in F (V) such that G <= F and let < be a minima ordering for F. The map G is a separation of F if and only if, for each minimum X for F, we have F(X,<) = G(X,<).
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79. Theorem (reconstruction from ordered pass values)
Let F be in F (V) and let < be a minima ordering for F. Let T be a tree associated to F(X,<). The pass values between all minima of F may be reconstructed from T.
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80. Ordered dynamics
The notion of ordered pass values leads to a new definition of the dynamics of a minimum: Dyn(X; F, <) = F(X, <) – F(X)
This new definition of dynamics fully agrees with the notion of separation.
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81. Segmentation
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82. ISMM 2005

83. Watershed
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84. Segmentation based on dynamics
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85. Segmentation based on dynamics
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86. Minima ordering 10 9 8 6 7 5 2 1 3 4
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87. Dynamics 10 9 8 6 7 5 2 1 3 4
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88. Dynamics 10 9 8 6 7 5 2 1 3 4
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89. Dynamics 6 1 3
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90. Dynamics 6 1 3
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91. Geodesic reconstruction6 1 3
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92. Watershed 6 1 3
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93. ISMM 2005

94. ISMM 2005

95. Watershed
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96. ISMM 2005

97. Dyn > 9
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98. Dyn > 9
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99. Dyn > 22
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100. Dyn > 22
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101. ‘Duality’
Let (V,E) be a connected graph and let E’ be a subset of E. We say that an edge u = {x,y} in E’ is destructible (for E’) if x and y belong to the same connected component of (V, E’\{u})
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102. Homotopy: an illustration
F(x,y)
G(x,y) F1 G1 x x
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103. Homotopy: an illustration
F(x,y)
G(x,y) x x F2 G2 F1 G1
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104. Watershed transform
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105. k-separation y x x and y are 8-separated ISMM 2005 40 40 40 40 40 401 1 2 3 10 5 25 5 4 4 4 40 x y 40 1 2 8 6 5 5 20 3 2 3 40 40 3 3 2 3 10 6 6 6 22 2 3 40 40 6 6 40 6 11 11 11 25 4 4 4 40 40 40 35 10 30 15 15 15 35 31 36 10 40 40 10 8 5 10 32 33 34 10 10 15 38 40 40 8 5 1 1 15 40 10 6 3 15 20 40 40 10 8 5 10 15 35 15 6 6 15 35 40 40 40 40 40 40 40 40 40 40 40 40 40 40
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106. x and y are NOT separated (they are linked)
k-separation
x and y are NOT separated (they are linked) 40 40 40 40 40 40 40 40 40 40 40 40 40 40 1 1 2 3 10 5 25 5 4 4 4 40 40 1 2 8 6 5 5 20 3 2 3 40 x y 40 3 3 2 3 10 6 6 6 22 2 3 40 40 6 6 40 6 11 11 11 25 4 4 4 40 40 40 35 10 30 15 15 15 35 31 36 10 40 40 10 8 5 10 32 33 34 10 10 15 38 40 40 8 5 1 1 15 40 10 6 3 15 20 40 40 10 8 5 10 15 35 15 6 6 15 35 40 40 40 40 40 40 40 40 40 40 40 40 40 40
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107. Pass value 20 40 40 40 40 40 40 40 40 40 40 40 40 40 40 1 1 2 8 3 10 5 25 5 20 4 4 4 40 40 1 2 8 6 5 5 20 3 2 3 40 40 3 3 2 3 10 6 6 6 22 2 3 40 30 31 40 6 6 40 6 11 11 11 25 4 4 4 40 31 30 40 40 35 10 30 15 15 15 35 31 36 10 40 40 10 8 5 10 32 33 34 10 10 15 38 40 40 8 5 1 1 15 40 10 6 3 15 20 40 40 10 8 5 10 15 35 15 6 6 15 35 40 40 40 40 40 40 40 40 40 40 40 40 40 40
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108. ISMM 2005

109. ISMM 2005

110. ISMM 2005

111. Cross-sections, components
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112. Cross-sections, components
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